Description
This is a presentation describes bond price volatility talks about characteristics, option free bond, measures, modified duration, portfolio, price yield relation, convexity etc
Bond Price Volatility
Price Volatility Characteristics
Exhibit 4-3. Instantaneous Percentage Price Change for 6 Hypothetical Bonds Six hypothetical bonds, priced initially to yield 9%: 9% coupon, 5 years to maturity, price = $100.0000 9% coupon, 25 years to maturity, price = 100.0000 6% coupon, 5 years to maturity, price = 88.1309 6% coupon, 25 years to maturity, price = 70.3570 0% coupon, 5 years to maturity, price = 64.3928 0% coupon, 25 years to maturity, price = 11.0710 Yield Change to: 6 7 8 8.5 8.9 8.99 9.01 9.1 9.5 10 11 12 Change Percentage Price Change (coupon/maturity in years) in BP 9%/5 9%/25 6%/5 6%/25 0%/5 0%/25 –300 12.8 38.59 13.47 42.13 15.56 106.04 –200 8.32 23.46 8.75 25.46 10.09 61.73 –100 4.06 10.74 4.26 11.6 4.91 27.1 –50 2 5.15 2.11 5.55 2.42 12.72 –10 0.4 1 0.42 1.07 0.48 2.42 –1 0.04 0.1 0.04 0.11 0.05 0.24 1 –0.04 –0.10 –0.04 –0.11 –0.05 –0.24 10 –0.39 –0.98 –0.41 –1.05 –0.48 –2.36 50 –1.95 –4.75 –2.05 –5.09 –2.36 –11.26 100 –3.86 –9.13 –4.06 –9.76 –4.66 –21.23 200 –7.54 –16.93 –7.91 –18.03 –9.08 –37.89 300 –11.04 –23.64 –11.59 –25.08 –13.28 –50.96
Price Volatility of Option-Free Bond
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?
?
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Although the prices of all option-free bonds move in opposite direction from the change in yield required, the % price change is not the same for all bonds. For very small changes in the yield required, the % price change for a given bond is roughly the same, whether the yield required increases or decreases. For large changes in the required yield, the % price change is not the same for an increase in the required yield as it is for a decrease in the required yield. For a given large change in basis points, the % price increase is greater than the % price decrease.
Price Volatility
?
?
?
For a given term to maturity and initial yield, the price volatility of a bond is greater, the lower the coupon rate. For a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility. The higher the YTM at which a bond trades, the lower the price volatility.
Measures of Price Volatility
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price value of a basis point – gives dollar price volatility not %
Bond 5-year 9% coupon 25-year 9% coupon 5-year 6% coupon 25-year 6% coupon 5-year zero-coupon 25-year zeroInitial Price (9% Yield) 100 100 88.1309 70.357 64.3928 11.071 Price at 9.01% 99.9604 99.9013 88.0945 70.2824 64.362 11.0445 Price Value of a BP 0.0396 0.0987 0.0364 0.0746 0.0308 0.0265
?
?
yield value of a price change duration
Duration
Duration
Exhibit 4-5. Calculation of Macaulay Duration and Modified Duration Coupon rate: 9.00% Term (years): 5 Initial yield: 9.00% Period, t CF PV of $1 at 4.5% PV of CF t x PVCF 1 $4.50 0.956937 4.30622 4.30622 2 4.50 0.915729 4.120785 8.24156 3 4.50 0.876296 3.943335 11.83 4 4.50 0.838561 3.773526 15.0941 5 4.50 0.802451 3.61103 18.05514 6 4.50 0.767895 3.455531 20.73318 7 4.50 0.734828 3.306728 23.14709 8 4.50 0.703185 3.164333 25.31466 9 4.50 0.672904 3.02807 27.25262 10 104.50 0.643927 67.290443 672.90442 100 826.87899
Duration
Exhibit 4-6. Calculation of Macaulay Duration and Modified Duration Coupon rate: 6.00% Term (years): 5 Initial yield: 9.00% Period, t CF PV of $1 at 4.5% PV of CF 1 $3.00 0.956937 2.870813 2 3.00 0.915729 2.74719 3 3.00 0.876296 2.62889 4 3.00 0.838561 2.515684 5 3.00 0.802451 2.407353 6 3.00 0.767895 2.303687 7 3.00 0.734828 2.204485 8 3.00 0.703185 2.109555 9 3.00 0.672904 2.018713 10 103.00 0.643927 66.324551 88.130923
t x PVCF 2.87081 5.49437 7.88666 10.06273 12.03676 13.82212 15.43139 16.87644 18.16841 663.24551 765.8952
Modified Duration
Duration
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duration is less than (coupon bond) or equal to (zero coupon bond) the term to maturity all else equal,
? ? ?
the lower the coupon, the larger the duration the longer the maturity, the larger the duration the lower the yield, the larger the duration
?
the longer the duration, the greater the price volatility
Duration
?
?
dollar duration = (-MD) * P spread duration – measure of how a non-Treasury bond’s price will change if the spread sought by the market changes
?
?
?
spread duration = 0 for Treasury for fixed rate security it is the approximate change in the price of a fixed-rate bond for a 100 bp change in the spread for a floater, a spread duration of 1.4 means that if the spread the market requires changes by 100 bp, the floater’s price will change by about 1.4%
?
portfolio duration – weighted average of bonds’ durations
Portfolio Duration
Bond 10% 5yr 8% 15yr 14% 30yr
Bond 10% 5yr 8% 15yr 14% 30yr
Par Amt Owned $4 million $5 million $1 million
Price ($) 100.0000 84.6275 137.8586
Market Value $4,000,000 $4,231,375 $1,378,586
Yield (%) 10 10 10 Duration 3.861 8.047 9.168
Portfolio Duration
Bond 10% 5yr 8% 15yr 14% 30yr Market Value $4,000,000 $4,231,375 $1,378,586 Duration Change in Value for 50bp Change in Yield 3.861 $77,220 8.047 170,249 9.168 63,194 Total $310,663
Measures of Bond Price Volatility
Price-Yield Relationship
Price Approximation using Duration
Convexity
?
second derivative of price-yield is dollar convexity measure of bond convexity measure convexity measure in terms of periods squared so to convert to annual figure, divide by 4
?
?
Calculation of Convexity for 5 Year, 9%, Selling to Yield 9% (Price = 100)
Period, t CF t(t + 1)CF
1 2 3 4 5 6 7 8 9 10
$4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $104.50
0.876296 0.838561 0.802451 0.767895 0.734828 0.703185 0.672904 0.643927 0.616198 0.589663
9 27 54 90 135 189 252 324 405 11,495 12,980
7.886 22.641 43.332 69.11 99.201 132.901 169.571 208.632 249.56 6,778.19 7,781.02
Calculation of Convexity for 5 Yr, 6%, Selling to Yield 9% (P=88.1309)
Period, t CF t(t + 1)CF
1 2 3 4 5 6 7 8 9 10
$3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $103.00
0.876296 0.838561 0.802451 0.767895 0.734828 0.703185 0.672904 0.643927 0.616198 0.589663
6 18 36 60 90 126 168 216 270 11,330 12,320
5.257 15.094 28.888 46.073 66.134 88.601 113.047 139.088 166.373 6,680.89 7,349.45
Convexity
consider the 25-year 6% bond selling at 70.357 to yield 9%
% Price Change
?
consider a 25 year 6% bond selling to yield 9%
? ?
MD = 10.62, convexity = 182.92 required yield increases 200 bp from 9% to 11%
?
estimated price change due to duration and convexity is 21.24% + 3.66% = -17.58%
Convexity
?
?
implication of convexity for bonds when yields change market takes convexity into account when pricing bonds
?
but to what extent should there be difference?
Convexity
1.
2.
3.
As the required yield increases (decreases), the convexity of a bond decreases (increases). This property is referred to as positive convexity. For a given yield and maturity, the lower the coupon, the greater the convexity of a bond. For a given yield and modified duration, the lower the coupon, the smaller the convexity.
Approximating Duration
1.
Use the 25 year, 6% bond trading at 9%. Increase the yield by 10bp from 9% to 9.1%. So ?y = 0.001.
The new price is P+ = 69.6164.
2.
3.
Decrease the yield on the bond by 10 bp from 9% to 8.9%. The new price is P- = 71.1105. Because the initial price, P0, is 70.3570, the duration can be approximated as follows
Approximating Duration
1.
2.
3.
Increase the yield on the bond by a small number of bp and determine the new price at this higher yield level. New price is P+. Decrease the yield on the bond by the same number of bp and calculate the new price. PLetting P0 be the initial price, duration can be approximated using the following where ?y is the
change in yield used to calculate the new prices. This gives the average % price change relative to the initial price per 1-bp change in yield.
Approximating Convexity
doc_786910771.ppt
This is a presentation describes bond price volatility talks about characteristics, option free bond, measures, modified duration, portfolio, price yield relation, convexity etc
Bond Price Volatility
Price Volatility Characteristics
Exhibit 4-3. Instantaneous Percentage Price Change for 6 Hypothetical Bonds Six hypothetical bonds, priced initially to yield 9%: 9% coupon, 5 years to maturity, price = $100.0000 9% coupon, 25 years to maturity, price = 100.0000 6% coupon, 5 years to maturity, price = 88.1309 6% coupon, 25 years to maturity, price = 70.3570 0% coupon, 5 years to maturity, price = 64.3928 0% coupon, 25 years to maturity, price = 11.0710 Yield Change to: 6 7 8 8.5 8.9 8.99 9.01 9.1 9.5 10 11 12 Change Percentage Price Change (coupon/maturity in years) in BP 9%/5 9%/25 6%/5 6%/25 0%/5 0%/25 –300 12.8 38.59 13.47 42.13 15.56 106.04 –200 8.32 23.46 8.75 25.46 10.09 61.73 –100 4.06 10.74 4.26 11.6 4.91 27.1 –50 2 5.15 2.11 5.55 2.42 12.72 –10 0.4 1 0.42 1.07 0.48 2.42 –1 0.04 0.1 0.04 0.11 0.05 0.24 1 –0.04 –0.10 –0.04 –0.11 –0.05 –0.24 10 –0.39 –0.98 –0.41 –1.05 –0.48 –2.36 50 –1.95 –4.75 –2.05 –5.09 –2.36 –11.26 100 –3.86 –9.13 –4.06 –9.76 –4.66 –21.23 200 –7.54 –16.93 –7.91 –18.03 –9.08 –37.89 300 –11.04 –23.64 –11.59 –25.08 –13.28 –50.96
Price Volatility of Option-Free Bond
?
?
?
?
Although the prices of all option-free bonds move in opposite direction from the change in yield required, the % price change is not the same for all bonds. For very small changes in the yield required, the % price change for a given bond is roughly the same, whether the yield required increases or decreases. For large changes in the required yield, the % price change is not the same for an increase in the required yield as it is for a decrease in the required yield. For a given large change in basis points, the % price increase is greater than the % price decrease.
Price Volatility
?
?
?
For a given term to maturity and initial yield, the price volatility of a bond is greater, the lower the coupon rate. For a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility. The higher the YTM at which a bond trades, the lower the price volatility.
Measures of Price Volatility
?
price value of a basis point – gives dollar price volatility not %
Bond 5-year 9% coupon 25-year 9% coupon 5-year 6% coupon 25-year 6% coupon 5-year zero-coupon 25-year zeroInitial Price (9% Yield) 100 100 88.1309 70.357 64.3928 11.071 Price at 9.01% 99.9604 99.9013 88.0945 70.2824 64.362 11.0445 Price Value of a BP 0.0396 0.0987 0.0364 0.0746 0.0308 0.0265
?
?
yield value of a price change duration
Duration
Duration
Exhibit 4-5. Calculation of Macaulay Duration and Modified Duration Coupon rate: 9.00% Term (years): 5 Initial yield: 9.00% Period, t CF PV of $1 at 4.5% PV of CF t x PVCF 1 $4.50 0.956937 4.30622 4.30622 2 4.50 0.915729 4.120785 8.24156 3 4.50 0.876296 3.943335 11.83 4 4.50 0.838561 3.773526 15.0941 5 4.50 0.802451 3.61103 18.05514 6 4.50 0.767895 3.455531 20.73318 7 4.50 0.734828 3.306728 23.14709 8 4.50 0.703185 3.164333 25.31466 9 4.50 0.672904 3.02807 27.25262 10 104.50 0.643927 67.290443 672.90442 100 826.87899
Duration
Exhibit 4-6. Calculation of Macaulay Duration and Modified Duration Coupon rate: 6.00% Term (years): 5 Initial yield: 9.00% Period, t CF PV of $1 at 4.5% PV of CF 1 $3.00 0.956937 2.870813 2 3.00 0.915729 2.74719 3 3.00 0.876296 2.62889 4 3.00 0.838561 2.515684 5 3.00 0.802451 2.407353 6 3.00 0.767895 2.303687 7 3.00 0.734828 2.204485 8 3.00 0.703185 2.109555 9 3.00 0.672904 2.018713 10 103.00 0.643927 66.324551 88.130923
t x PVCF 2.87081 5.49437 7.88666 10.06273 12.03676 13.82212 15.43139 16.87644 18.16841 663.24551 765.8952
Modified Duration
Duration
?
?
duration is less than (coupon bond) or equal to (zero coupon bond) the term to maturity all else equal,
? ? ?
the lower the coupon, the larger the duration the longer the maturity, the larger the duration the lower the yield, the larger the duration
?
the longer the duration, the greater the price volatility
Duration
?
?
dollar duration = (-MD) * P spread duration – measure of how a non-Treasury bond’s price will change if the spread sought by the market changes
?
?
?
spread duration = 0 for Treasury for fixed rate security it is the approximate change in the price of a fixed-rate bond for a 100 bp change in the spread for a floater, a spread duration of 1.4 means that if the spread the market requires changes by 100 bp, the floater’s price will change by about 1.4%
?
portfolio duration – weighted average of bonds’ durations
Portfolio Duration
Bond 10% 5yr 8% 15yr 14% 30yr
Bond 10% 5yr 8% 15yr 14% 30yr
Par Amt Owned $4 million $5 million $1 million
Price ($) 100.0000 84.6275 137.8586
Market Value $4,000,000 $4,231,375 $1,378,586
Yield (%) 10 10 10 Duration 3.861 8.047 9.168
Portfolio Duration
Bond 10% 5yr 8% 15yr 14% 30yr Market Value $4,000,000 $4,231,375 $1,378,586 Duration Change in Value for 50bp Change in Yield 3.861 $77,220 8.047 170,249 9.168 63,194 Total $310,663
Measures of Bond Price Volatility
Price-Yield Relationship
Price Approximation using Duration
Convexity
?
second derivative of price-yield is dollar convexity measure of bond convexity measure convexity measure in terms of periods squared so to convert to annual figure, divide by 4
?
?
Calculation of Convexity for 5 Year, 9%, Selling to Yield 9% (Price = 100)
Period, t CF t(t + 1)CF
1 2 3 4 5 6 7 8 9 10
$4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $4.50 $104.50
0.876296 0.838561 0.802451 0.767895 0.734828 0.703185 0.672904 0.643927 0.616198 0.589663
9 27 54 90 135 189 252 324 405 11,495 12,980
7.886 22.641 43.332 69.11 99.201 132.901 169.571 208.632 249.56 6,778.19 7,781.02
Calculation of Convexity for 5 Yr, 6%, Selling to Yield 9% (P=88.1309)
Period, t CF t(t + 1)CF
1 2 3 4 5 6 7 8 9 10
$3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $3.00 $103.00
0.876296 0.838561 0.802451 0.767895 0.734828 0.703185 0.672904 0.643927 0.616198 0.589663
6 18 36 60 90 126 168 216 270 11,330 12,320
5.257 15.094 28.888 46.073 66.134 88.601 113.047 139.088 166.373 6,680.89 7,349.45
Convexity
consider the 25-year 6% bond selling at 70.357 to yield 9%
% Price Change
?
consider a 25 year 6% bond selling to yield 9%
? ?
MD = 10.62, convexity = 182.92 required yield increases 200 bp from 9% to 11%
?
estimated price change due to duration and convexity is 21.24% + 3.66% = -17.58%
Convexity
?
?
implication of convexity for bonds when yields change market takes convexity into account when pricing bonds
?
but to what extent should there be difference?
Convexity
1.
2.
3.
As the required yield increases (decreases), the convexity of a bond decreases (increases). This property is referred to as positive convexity. For a given yield and maturity, the lower the coupon, the greater the convexity of a bond. For a given yield and modified duration, the lower the coupon, the smaller the convexity.
Approximating Duration
1.
Use the 25 year, 6% bond trading at 9%. Increase the yield by 10bp from 9% to 9.1%. So ?y = 0.001.
The new price is P+ = 69.6164.
2.
3.
Decrease the yield on the bond by 10 bp from 9% to 8.9%. The new price is P- = 71.1105. Because the initial price, P0, is 70.3570, the duration can be approximated as follows
Approximating Duration
1.
2.
3.
Increase the yield on the bond by a small number of bp and determine the new price at this higher yield level. New price is P+. Decrease the yield on the bond by the same number of bp and calculate the new price. PLetting P0 be the initial price, duration can be approximated using the following where ?y is the
change in yield used to calculate the new prices. This gives the average % price change relative to the initial price per 1-bp change in yield.
Approximating Convexity
doc_786910771.ppt